Explain how to leverage the WordPress REST API to build a headless application, and describe how to retrieve and manipulate data.

Leveraging the WordPress REST API to build a headless application involves decoupling the front-end presentation layer from the back-end data management system. In a traditional WordPress site, the front-end (themes) and the back-end (WordPress core) are tightly coupled. However, with a headless approach, WordPress becomes solely a content repository, with the REST API serving as the primary mechanism for interacting with and retrieving data for a completely separate front-end application, which can be built using frameworks like React, Angular, or Vue.js, or even a mobile application.

To begin using the REST API, we first need to understand the basic structure of API endpoints. Each endpoint corresponds to a particular type of data or action. For example, the default endpoint for posts is `/wp-json/wp/v2/posts`. To retrieve all posts from your WordPress site, you would make a GET request to this endpoint. A typical request might look like this: `https://example.com/wp-json/wp/v2/posts`.

The response from this request is a JSON object containing an array of posts. Each post object includes various fields such as `id`, `title`, `content`, `date`, `author`, and `categories`. Using JavaScript on the front-end, you can make an HTTP request using `fetch` or `axios` to retrieve the data from this API endpoint and then process this JSON data to display on your front end application. Here’s a simple example of fetching posts in JavaScript using fetch:

```javascript
fetch('https://example.com/wp-json/wp/v2/posts')
.then(response => response.json())
.then(posts => {
posts.forEach(post => {
console.log("Title: " + post.title.rendered);
console.log("Content: " + post.content.rendered);

});
})
.catch(error => console.error('Error fetching posts:', error));
```
This code fetches the posts data from the api and then logs the post title and content in the console.

To retrieve a specific post instead of a list of posts, you would add the post ID to the end of the posts endpoint, like `https://example.com/wp-json/wp/v2/posts/123`, where 123 is the ID of the post you want. This will give the JSON data of that particular post.

Beyond retrieving posts, you can also use the API to interact with other types of data, including pages, categories, tags, users, and media. For example, to fetch categories, you would use the endpoint `/wp-json/wp/v2/categories`. To retrieve users it would be `/wp-json/wp/v2/users`. Custom post types created in WordPress will also have their dedicated API endpoints available at `/wp-json/wp/v2/{custom-post-type-name}`, so if you have a custom post type called 'books' it will be `/wp-json/wp/v2/books`.

Manipulating data via the REST API is done primarily using POST, PUT, and DELETE requests. For instance, to create a new post, you would make a POST request to the `/wp-json/wp/v2/posts` endpoint. You would also need to provide the post data in the body of the request as a JSON object, including at least the `title` and `content`. You also have to authenticate the request before posting new data to the server. You can do this by passing in the user’s credentials via HTTP headers. This process typically requires using an authentication mechanism like Basic Auth, JWT authentication, or OAuth 2.0 as the WordPress REST API does not store authentication details in cookies.

Updating an existing post would involve a PUT request to the same endpoint, including the post ID in the URL and updated data in the request body, example `/wp-json/wp/v2/posts/123`. Likewise, deleting a post is done through a DELETE request to the post's specific endpoint `/wp-json/wp/v2/posts/123`.

Using the REST API to build a headless app is advantageous because it provides flexibility and scalability by decoupling the front-end from the back-end. It allows you to leverage the power of WordPress as a robust CMS while building rich and interactive user experiences with modern JavaScript front-end frameworks or even mobile applications. This approach also enhances performance by allowing you to optimize the front-end separately from WordPress and also offers more customization options and creative freedom by not being limited by traditional WordPress theme development. The WordPress REST API is a very important tool that allows developers to build very modern and powerful applications.